ABSTRACT
Whereas differential equations relate a function to its derivatives, a difference equation will be a relationship on a sequence of numbers. We have already seen difference equations in sections 4.3 and 4.4, when we found a recursion relationship for the coefficients of a series. This recursion formula is an example of a difference equation on the sequence of coefficients. Sometimes we were able to solve the recursion relationship, which allowed us to express the nth coefficient as a function of n, and hence can express the solution to the differential equation as an infinite sum. But there are also times where the recursion relationship cannot be solved in closed form. Nonetheless, we will be able to use asymptotics on this recursion relationship to extract important information from the sequence, such as the radius of convergence. This chapter will focus on the rich topic of difference equations, their uses, and how to find the asymptotic approximations to the solutions.
